Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 223
The topological entropy of g is the entropy on X and is denoted by ht(g). The reader
can check that if g is an isometry, then ht(g)=0. In complex dynamics, we often
have that for ε small enough, 1
n logN(X,n,ε) converge to ht(g).
Let f be an endomorphism of algebraic degree d ≥ 2ofPkas above. As we
have seen, the iterate f n of f has algebraic degree dn .IfZ is an algebraic set
in Pk of codimension p then the degree of f −n (Z), counted with multiplicity, is
equal to d pn deg(Z) and the degree of f n (Z), counting with multiplicity, is equal to
d (k−p)n deg(Z). This is a consequence of Bézout’s theorem. Recall that the degree
of an algebraic set of codimension p in Pk is the number of points of intersection
with a generic projective subspace of dimension p.
The pull-back by f induces a linear map f ∗ : H p,p (Pk ,C) → H p,p (Pk ,C) which
is just the multiplication by d p . The constant d p is the dynamical degree of order
p of f . Dynamical degrees were considered by Gromov in [GR] where he
introduced a method to bound the topological entropy from above. We will see
that they measure the volume growth of the graphs. The degree of maximal order
dk is also called the topological degree. It is equal to the number of points in a
fiber counting with multiplicity. The push-forward by f n induces a linear map
f∗ : H p,p (Pk ,C) → H p,p (Pk ,C) which is the multiplication by dk−p . These operations
act continuously on positive closed currents and hence, the actions are
compatible with cohomology, see Appendix A.1.
We have the following result due to Gromov [GR] for the upper bound and to
Misiurewicz-Przytycky [MP] for the lower bound of the entropy.
Theorem 1.108. Let f be a holomorphic endomorphism of algebraic degree d on
P k . Then the topological entropy ht( f ) of f is equal to k logd, i.e. to the logarithm
of the maximal dynamical degree.
The inequality ht( f ) ≥ k logd is a consequence of the following result which is
valid for arbitrary C 1 maps [MP].
Theorem 1.109 (Misuriewicz-Przytycki). Let X be a compact smooth orientable
manifold and g : X → XaC 1 map. Then
ht(g) ≥ log|deg(g)|.
Recall that the degree of g is defined as follows. Let Ω be a continuous form of
maximal degree on X such that �
X Ω �=0. Then
deg(g) :=
�
X g∗ (Ω)
�
X
Ω ·
The number is independent of the choice of Ω. WhenX is a complex manifold,
it is necessarily orientable and deg(g) is just the generic number of preimages of
a point, i.e. the topological degree of g. In our case, the topological degree of f is
equal to d k .So,ht( f ) ≥ k logd.
Instead of using Misuriewicz-Przytycki theorem, it is also possible to apply the
following important result due to Yomdin [YO].